where \(D_{1}\)  and \({D_{2}}\)  are two bounded open simply-connected domains with \(\overline{D_{2}}\subset D_{1}. \)  This means that the boundary of \(D_{0}\)  is given by two interfaces, one of them is contained in the open region delimited by the second one.
According to the structure of \((QGSW)_{\lambda }\) , every radial initial domain \(D_0\)  generates a trivial stationary solution, and therefore a V-state rotating with any angular velocity. Basic examples are given by the discs in the simply-connected case or the annuli in the doubly-connected case.
The first non-trivial examples of uniformly rotating solutions for Euler equations are Kirchhoff ellipses which rotate with the angular velocity \(\Omega =\frac{ab}{(a+b)^{2}}\)  where \(a\)  and \(b\)  are the semi-axes of the ellipse (see [1]} and [2]}). In [3]} Deem and Zabusky established numerically the existence of simply-connected rotating patches with \(\mathbf {m}\) -fold symmetry for \(\mathbf {m}>2.\)  An analytical proof based on bifurcation theory and complex analysis tools was performed by Burbea in [4]} showing the existence of \(\mathbf {m}\) -fold (for any \(\mathbf {m}\in \mathbb {N}^{*}\) ) symmetric V-states bifurcating from Rankine vortices with angular velocity \(\Omega _{\mathbf {m}}:=\frac{\mathbf {m}-1}{2\mathbf {m}}\) .
In the spirit of Burbea's work, a lot of results on \(\mathbf {m}\) -fold V-states have been obtained both for simply and doubly-connected cases for Euler, \((SQG)_{\alpha }\)  and \((QGSW)_{\lambda }\)  equations in the past decade. We may refer to [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}. From this long list, we shall make some comments on two contributions from [6]}, [11]} related to the current work. In [11]}, the authors proved for Euler equations that under the condition
\(1+b^{\mathbf {m}}-\frac{\mathbf {m}(1-b^{2})}{2}<0,\) 
